Asymptote A

Percentage Accurate: 77.9% → 99.9%

Time: 5.8s

Alternatives: 4

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) - (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 1} - \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) - (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
}

Python

def code(x):
	return (1.0 / (x + 1.0)) - (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x + 1.0)) - (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(-1 \cdot x\_m\right) \cdot x\_m\\ t_1 := {\left(-1 \cdot t\_0\right)}^{2}\\ \mathbf{if}\;x\_m \leq 200000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m - 1, 1, \mathsf{fma}\left(-1 \cdot x\_m, 1, -1\right)\right)}{\mathsf{fma}\left(t\_0, -1, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({t\_1}^{-0.5} \cdot \left(--2\right), -1, \frac{-2}{t\_1}\right), 1, -2\right)}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (* -1.0 x_m) x_m)) (t_1 (pow (* -1.0 t_0) 2.0)))
   (if (<= x_m 200000000.0)
     (/ (fma (- x_m 1.0) 1.0 (fma (* -1.0 x_m) 1.0 -1.0)) (fma t_0 -1.0 -1.0))
     (/
      (/
       (fma (fma (* (pow t_1 -0.5) (- -2.0)) -1.0 (/ -2.0 t_1)) 1.0 -2.0)
       x_m)
      x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (-1.0 * x_m) * x_m;
	double t_1 = pow((-1.0 * t_0), 2.0);
	double tmp;
	if (x_m <= 200000000.0) {
		tmp = fma((x_m - 1.0), 1.0, fma((-1.0 * x_m), 1.0, -1.0)) / fma(t_0, -1.0, -1.0);
	} else {
		tmp = (fma(fma((pow(t_1, -0.5) * -(-2.0)), -1.0, (-2.0 / t_1)), 1.0, -2.0) / x_m) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(-1.0 * x_m) * x_m)
	t_1 = Float64(-1.0 * t_0) ^ 2.0
	tmp = 0.0
	if (x_m <= 200000000.0)
		tmp = Float64(fma(Float64(x_m - 1.0), 1.0, fma(Float64(-1.0 * x_m), 1.0, -1.0)) / fma(t_0, -1.0, -1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64((t_1 ^ -0.5) * Float64(-(-2.0))), -1.0, Float64(-2.0 / t_1)), 1.0, -2.0) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(-1.0 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(-1.0 * t$95$0), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x$95$m, 200000000.0], N[(N[(N[(x$95$m - 1.0), $MachinePrecision] * 1.0 + N[(N[(-1.0 * x$95$m), $MachinePrecision] * 1.0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * -1.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[t$95$1, -0.5], $MachinePrecision] * (--2.0)), $MachinePrecision] * -1.0 + N[(-2.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * 1.0 + -2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e8

   1. Initial program 86.1%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1} - \frac{1}{x - 1}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x + 1}} - \frac{1}{x - 1} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{x + 1}} - \frac{1}{x - 1} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1}{x + 1} - \frac{1}{\color{blue}{x - 1}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1}{x + 1} - \color{blue}{\frac{1}{x - 1}} \]

      6. frac-sub N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]

      7. difference-of-squares-rev N/A

         \[\leadsto \frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{\color{blue}{x \cdot x - 1 \cdot 1}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - 1\right) - \left(x + 1\right) \cdot 1}{x \cdot x - 1 \cdot 1}} \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right) \cdot 1}}{x \cdot x - 1 \cdot 1} \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x - 1\right) \cdot 1} + \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right) \cdot 1}{x \cdot x - 1 \cdot 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x - 1, 1, \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right) \cdot 1\right)}}{x \cdot x - 1 \cdot 1} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x - 1}, 1, \left(\mathsf{neg}\left(\left(x + 1\right)\right)\right) \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \color{blue}{\left(\mathsf{neg}\left(\left(x + 1\right)\right)\right) \cdot 1}\right)}{x \cdot x - 1 \cdot 1} \]

      14. distribute-neg-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{-1}\right) \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      16. *-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot 1} + -1\right) \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), 1, -1\right)} \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      18. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \mathsf{fma}\left(\color{blue}{-1 \cdot x}, 1, -1\right) \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \mathsf{fma}\left(\color{blue}{-1 \cdot x}, 1, -1\right) \cdot 1\right)}{x \cdot x - 1 \cdot 1} \]

      20. difference-of-squares-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \mathsf{fma}\left(-1 \cdot x, 1, -1\right) \cdot 1\right)}{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]

      21. difference-of-sqr--1-rev N/A

          \[\leadsto \frac{\mathsf{fma}\left(x - 1, 1, \mathsf{fma}\left(-1 \cdot x, 1, -1\right) \cdot 1\right)}{\color{blue}{x \cdot x + -1}} \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x - 1, 1, \mathsf{fma}\left(-1 \cdot x, 1, -1\right) \cdot 1\right)}{\mathsf{fma}\left(\left(-1 \cdot x\right) \cdot x, -1, -1\right)}} \]

   if 2e8 < x

   1. Initial program 49.6%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{2}} - \color{blue}{\frac{2}{{x}^{2}}} \]

      2. frac-sub N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{{x}^{\color{blue}{\left(2 + 2\right)}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{{x}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{\color{blue}{{x}^{4}}} \]

   5. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1} \cdot 2, 1, 2\right)}{x}\right) \cdot \left(x \cdot x\right) - \left(x \cdot x\right) \cdot 2}{{\left(x \cdot x\right)}^{2}}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left({\left({\left(x \cdot x\right)}^{2}\right)}^{-0.5} \cdot -1\right) \cdot -2, -1, \frac{-2}{{\left(x \cdot x\right)}^{2}}\right), 1, -2\right)}{-1 \cdot x}}{\color{blue}{-1 \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x - 1, 1, \mathsf{fma}\left(-1 \cdot x, 1, -1\right)\right)}{\mathsf{fma}\left(\left(-1 \cdot x\right) \cdot x, -1, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({\left({\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{2}\right)}^{-0.5} \cdot \left(--2\right), -1, \frac{-2}{{\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{2}}\right), 1, -2\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(-1 \cdot \left(\left(-1 \cdot x\_m\right) \cdot x\_m\right)\right)}^{2}\\ \mathbf{if}\;x\_m \leq 230:\\ \;\;\;\;\frac{1}{x\_m - -1} - \frac{1}{x\_m - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({t\_0}^{-0.5} \cdot \left(--2\right), -1, \frac{-2}{t\_0}\right), 1, -2\right)}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (* -1.0 (* (* -1.0 x_m) x_m)) 2.0)))
   (if (<= x_m 230.0)
     (- (/ 1.0 (- x_m -1.0)) (/ 1.0 (- x_m 1.0)))
     (/
      (/
       (fma (fma (* (pow t_0 -0.5) (- -2.0)) -1.0 (/ -2.0 t_0)) 1.0 -2.0)
       x_m)
      x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow((-1.0 * ((-1.0 * x_m) * x_m)), 2.0);
	double tmp;
	if (x_m <= 230.0) {
		tmp = (1.0 / (x_m - -1.0)) - (1.0 / (x_m - 1.0));
	} else {
		tmp = (fma(fma((pow(t_0, -0.5) * -(-2.0)), -1.0, (-2.0 / t_0)), 1.0, -2.0) / x_m) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(-1.0 * Float64(Float64(-1.0 * x_m) * x_m)) ^ 2.0
	tmp = 0.0
	if (x_m <= 230.0)
		tmp = Float64(Float64(1.0 / Float64(x_m - -1.0)) - Float64(1.0 / Float64(x_m - 1.0)));
	else
		tmp = Float64(Float64(fma(fma(Float64((t_0 ^ -0.5) * Float64(-(-2.0))), -1.0, Float64(-2.0 / t_0)), 1.0, -2.0) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(-1.0 * N[(N[(-1.0 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[x$95$m, 230.0], N[(N[(1.0 / N[(x$95$m - -1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[t$95$0, -0.5], $MachinePrecision] * (--2.0)), $MachinePrecision] * -1.0 + N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * 1.0 + -2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 230

   1. Initial program 86.3%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Add Preprocessing

   if 230 < x

   1. Initial program 50.1%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{2}} - \color{blue}{\frac{2}{{x}^{2}}} \]

      2. frac-sub N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{{x}^{\color{blue}{\left(2 + 2\right)}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{{x}^{4}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{\color{blue}{{x}^{4}}} \]

   5. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{\left(\frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1} \cdot 2, 1, 2\right)}{x}\right) \cdot \left(x \cdot x\right) - \left(x \cdot x\right) \cdot 2}{{\left(x \cdot x\right)}^{2}}} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left({\left({\left(x \cdot x\right)}^{2}\right)}^{-0.5} \cdot -1\right) \cdot -2, -1, \frac{-2}{{\left(x \cdot x\right)}^{2}}\right), 1, -2\right)}{-1 \cdot x}}{\color{blue}{-1 \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{1}{x - -1} - \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({\left({\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{2}\right)}^{-0.5} \cdot \left(--2\right), -1, \frac{-2}{{\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{2}}\right), 1, -2\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.3% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := {\left(-1 \cdot \left(\left(-1 \cdot x\_m\right) \cdot x\_m\right)\right)}^{2}\\ \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({t\_0}^{-0.5} \cdot \left(--2\right), -1, \frac{-2}{t\_0}\right), 1, -2\right)}{x\_m}}{x\_m} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (pow (* -1.0 (* (* -1.0 x_m) x_m)) 2.0)))
   (/
    (/ (fma (fma (* (pow t_0 -0.5) (- -2.0)) -1.0 (/ -2.0 t_0)) 1.0 -2.0) x_m)
    x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = pow((-1.0 * ((-1.0 * x_m) * x_m)), 2.0);
	return (fma(fma((pow(t_0, -0.5) * -(-2.0)), -1.0, (-2.0 / t_0)), 1.0, -2.0) / x_m) / x_m;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(-1.0 * Float64(Float64(-1.0 * x_m) * x_m)) ^ 2.0
	return Float64(Float64(fma(fma(Float64((t_0 ^ -0.5) * Float64(-(-2.0))), -1.0, Float64(-2.0 / t_0)), 1.0, -2.0) / x_m) / x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Power[N[(-1.0 * N[(N[(-1.0 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(N[(N[Power[t$95$0, -0.5], $MachinePrecision] * (--2.0)), $MachinePrecision] * -1.0 + N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * 1.0 + -2.0), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Initial program 76.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 2}{{x}^{2}}} \]
4. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto \frac{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}}{{x}^{2}} - \color{blue}{\frac{2}{{x}^{2}}} \]

   2. frac-sub N/A

      \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{{x}^{\color{blue}{\left(2 + 2\right)}}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{{x}^{4}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\left(-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot {x}^{2} - {x}^{2} \cdot 2}{\color{blue}{{x}^{4}}} \]

5. Applied rewrites 17.0%

   \[\leadsto \color{blue}{\frac{\left(\frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1} \cdot 2, 1, 2\right)}{x}\right) \cdot \left(x \cdot x\right) - \left(x \cdot x\right) \cdot 2}{{\left(x \cdot x\right)}^{2}}} \]

7. Applied rewrites 52.4%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left({\left({\left(x \cdot x\right)}^{2}\right)}^{-0.5} \cdot -1\right) \cdot -2, -1, \frac{-2}{{\left(x \cdot x\right)}^{2}}\right), 1, -2\right)}{-1 \cdot x}}{\color{blue}{-1 \cdot x}} \]

8. Final simplification 52.4%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({\left({\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{2}\right)}^{-0.5} \cdot \left(--2\right), -1, \frac{-2}{{\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{2}}\right), 1, -2\right)}{x}}{x} \]
9. Add Preprocessing

Alternative 4: 50.2% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{-1}{x\_m} \cdot \frac{\mathsf{fma}\left({\left(-1 \cdot \left(\left(-1 \cdot x\_m\right) \cdot x\_m\right)\right)}^{-1} \cdot 2, 1, 2\right)}{x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (/ -1.0 x_m)
  (/ (fma (* (pow (* -1.0 (* (* -1.0 x_m) x_m)) -1.0) 2.0) 1.0 2.0) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return (-1.0 / x_m) * (fma((pow((-1.0 * ((-1.0 * x_m) * x_m)), -1.0) * 2.0), 1.0, 2.0) / x_m);
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(-1.0 / x_m) * Float64(fma(Float64((Float64(-1.0 * Float64(Float64(-1.0 * x_m) * x_m)) ^ -1.0) * 2.0), 1.0, 2.0) / x_m))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-1.0 / x$95$m), $MachinePrecision] * N[(N[(N[(N[Power[N[(-1.0 * N[(N[(-1.0 * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision] * 1.0 + 2.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{{x}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)}{\color{blue}{{x}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{-1 \cdot \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)}{x \cdot \color{blue}{x}} \]

   3. times-frac N/A

      \[\leadsto \frac{-1}{x} \cdot \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{x}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{x} \cdot \color{blue}{\frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{x}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{-1}{x} \cdot \frac{\color{blue}{2 + 2 \cdot \frac{1}{{x}^{2}}}}{x} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{-1}{x} \cdot \frac{2 + 2 \cdot \frac{1}{{x}^{2}}}{\color{blue}{x}} \]

   7. +-commutative N/A

      \[\leadsto \frac{-1}{x} \cdot \frac{2 \cdot \frac{1}{{x}^{2}} + 2}{x} \]

   8. *-commutative N/A

      \[\leadsto \frac{-1}{x} \cdot \frac{\frac{1}{{x}^{2}} \cdot 2 + 2}{x} \]

   9. metadata-eval N/A

      \[\leadsto \frac{-1}{x} \cdot \frac{\frac{1}{{x}^{2}} \cdot \left(2 \cdot 1\right) + 2}{x} \]

   10. associate-*r* N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\left(\frac{1}{{x}^{2}} \cdot 2\right) \cdot 1 + 2}{x} \]

   11. *-commutative N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\left(2 \cdot \frac{1}{{x}^{2}}\right) \cdot 1 + 2}{x} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left(2 \cdot \frac{1}{{x}^{2}}, 1, 2\right)}{x} \]

   13. *-commutative N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left(\frac{1}{{x}^{2}} \cdot 2, 1, 2\right)}{x} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left(\frac{1}{{x}^{2}} \cdot 2, 1, 2\right)}{x} \]

   15. inv-pow N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left({x}^{2}\right)}^{-1} \cdot 2, 1, 2\right)}{x} \]

   16. lower-pow.f64 N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left({x}^{2}\right)}^{-1} \cdot 2, 1, 2\right)}{x} \]

   17. unpow2 N/A

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1} \cdot 2, 1, 2\right)}{x} \]

   18. lower-*.f64 52.2

       \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1} \cdot 2, 1, 2\right)}{x} \]

5. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{-1} \cdot 2, 1, 2\right)}{x}} \]

6. Final simplification 52.2%

   \[\leadsto \frac{-1}{x} \cdot \frac{\mathsf{fma}\left({\left(-1 \cdot \left(\left(-1 \cdot x\right) \cdot x\right)\right)}^{-1} \cdot 2, 1, 2\right)}{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))